Average Error: 0.4 → 0.7
Time: 24.1s
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[(\left(\frac{1}{6} - \left(\left(\pi \cdot u2\right) \cdot \left(\pi \cdot u2\right)\right) \cdot \frac{1}{3}\right) \cdot \left({\left({\left(\log u1\right)}^{1.0} \cdot {-2}^{1.0}\right)}^{0.5}\right) + 0.5)_*\]

Error

Bits error versus u1

Bits error versus u2

Derivation

  1. Initial program 0.4

    \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

  2. Initial simplification0.4

    \[\leadsto (\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right) \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right) + 0.5)_*\]

  3. Taylor expanded around 0 0.7

    \[\leadsto \color{blue}{\left(0.5 + \frac{1}{6} \cdot {\left({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{0.5}\right) - \frac{1}{3} \cdot \left(\left({u2}^{2} \cdot {\pi}^{2}\right) \cdot {\left({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{0.5}\right)}\]

  4. Simplified0.7

    \[\leadsto \color{blue}{(\left(\frac{1}{6} - \left(\left(u2 \cdot \pi\right) \cdot \left(u2 \cdot \pi\right)\right) \cdot \frac{1}{3}\right) \cdot \left({\left({\left(\log u1\right)}^{1.0} \cdot {-2}^{1.0}\right)}^{0.5}\right) + 0.5)_*}\]

  5. Final simplification0.7

    \[\leadsto (\left(\frac{1}{6} - \left(\left(\pi \cdot u2\right) \cdot \left(\pi \cdot u2\right)\right) \cdot \frac{1}{3}\right) \cdot \left({\left({\left(\log u1\right)}^{1.0} \cdot {-2}^{1.0}\right)}^{0.5}\right) + 0.5)_*\]

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0 u1 1) (<= 0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))

Details

Time bar (total: 18.6s)Debug log

start328.0ms

Algorithm
intervals

setup163.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.4b

localize60.0ms

Local error

Found 4 expressions with local error:

0.4b
(* (pow (* -2 (log u1)) 0.5) 1/6)
0.3b
(* PI u2)
0.1b
(pow (* -2 (log u1)) 0.5)
0.1b
(fma (cos (* 2 (* PI u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 47
Calls

4 calls. Slowest were:

10.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
2.0ms
(pow (* -2 (log u1)) 0.5)
0.0ms
(* PI u2)

series2.6s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.8s
(* (pow (* -2 (log u1)) 0.5) 1/6)
403.0ms
(pow (* -2 (log u1)) 0.5)
400.0ms
(fma (cos (* 2 (* PI u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)
56.0ms
(* PI u2)

simplify592.0ms

Counts
18 → 59
Calls

18 calls. Slowest were:

184.0ms
(- (+ 0.5 (* 1/6 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 0.5))) (* 1/3 (* (* (pow u2 2) (pow PI 2)) (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 0.5))))
105.0ms
(pow (* -2 (- (log -1) (log (/ -1 u1)))) 0.5)
87.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))

prune1.2s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize52.0ms

Local error

Found 4 expressions with local error:

0.4b
(fma (- 1/6 (* (* (* u2 PI) (* u2 PI)) 1/3)) (pow (* (pow (log u1) 1.0) (pow -2 1.0)) 0.5) 0.5)
0.3b
(* u2 PI)
0.3b
(* u2 PI)
0.3b
(* (* u2 PI) (* u2 PI))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

6.0ms
(* (* u2 PI) (* u2 PI))
0.0ms
(fma (- 1/6 (* (* (* u2 PI) (* u2 PI)) 1/3)) (pow (* (pow (log u1) 1.0) (pow -2 1.0)) 0.5) 0.5)
0.0ms
(* u2 PI)

series688.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

534.0ms
(fma (- 1/6 (* (* (* u2 PI) (* u2 PI)) 1/3)) (pow (* (pow (log u1) 1.0) (pow -2 1.0)) 0.5) 0.5)
80.0ms
(* (* u2 PI) (* u2 PI))
44.0ms
(* u2 PI)
30.0ms
(* u2 PI)

simplify1.3s

Counts
18 → 60
Calls

18 calls. Slowest were:

575.0ms
(* (* (* (* u2 PI) (* u2 PI)) (* u2 PI)) (* (* (* u2 PI) (* u2 PI)) (* u2 PI)))
249.0ms
(- (+ (* 1/6 (pow (* (pow -1 1.0) (* (pow -2 1.0) (pow (log (/ 1 u1)) 1.0))) 0.5)) 0.5) (* 1/3 (* (pow (* (pow (log (/ 1 u1)) 1.0) (* (pow -2 1.0) (pow -1 1.0))) 0.5) (* (pow u2 2) (pow PI 2)))))
192.0ms
(- (+ (* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5)) 0.5) (* 1/3 (* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (* (pow u2 2) (pow PI 2)))))

prune1.1s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

0.4b
(expm1 (log1p (* (pow (* -2 (log u1)) 0.5) 1/6)))
0.4b
(* (pow (* -2 (log u1)) 0.5) 1/6)
0.3b
(* PI u2)
0.2b
(log1p (* (pow (* -2 (log u1)) 0.5) 1/6))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
4 → 46
Calls

4 calls. Slowest were:

4.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
0.0ms
(log1p (* (pow (* -2 (log u1)) 0.5) 1/6))
0.0ms
(expm1 (log1p (* (pow (* -2 (log u1)) 0.5) 1/6)))

series2.9s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.1s
(log1p (* (pow (* -2 (log u1)) 0.5) 1/6))
975.0ms
(expm1 (log1p (* (pow (* -2 (log u1)) 0.5) 1/6)))
793.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
54.0ms
(* PI u2)

simplify340.0ms

Counts
17 → 58
Calls

17 calls. Slowest were:

83.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))
70.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))
58.0ms
(log (+ (* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5)) 1))

prune1.3s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 4 expressions with local error:

0.5b
(* (sqrt (* (pow (* -2 (log u1)) 0.5) 1/6)) (sqrt (* (pow (* -2 (log u1)) 0.5) 1/6)))
0.4b
(* (pow (* -2 (log u1)) 0.5) 1/6)
0.4b
(* (pow (* -2 (log u1)) 0.5) 1/6)
0.3b
(* PI u2)

rewrite45.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

28.0ms
(* (sqrt (* (pow (* -2 (log u1)) 0.5) 1/6)) (sqrt (* (pow (* -2 (log u1)) 0.5) 1/6)))
8.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
7.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)

series2.8s

Counts
4 → 12
Calls

4 calls. Slowest were:

938.0ms
(* (sqrt (* (pow (* -2 (log u1)) 0.5) 1/6)) (sqrt (* (pow (* -2 (log u1)) 0.5) 1/6)))
918.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
881.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
33.0ms
(* PI u2)

simplify652.0ms

Counts
36 → 78
Calls

36 calls. Slowest were:

128.0ms
(* (* (* (sqrt (* (pow (* -2 (log u1)) 0.5) 1/6)) (sqrt (* (pow (* -2 (log u1)) 0.5) 1/6))) (sqrt (* (pow (* -2 (log u1)) 0.5) 1/6))) (* (* (sqrt (* (pow (* -2 (log u1)) 0.5) 1/6)) (sqrt (* (pow (* -2 (log u1)) 0.5) 1/6))) (sqrt (* (pow (* -2 (log u1)) 0.5) 1/6))))
81.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))
72.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))

prune1.7s

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 0.0b

regimes635.0ms

Accuracy

0% (0.7b remaining)

Error of 0.7b against oracle of 0.0b and baseline of 0.7b

bsearch6.0ms